I am a bit worried that I sanded too aggressively--there are a few spots where the wood is noticeably thinner than it was previously (most worryingly at the root of the horizontal stabilizer), but not by much. Should I be concerned, or is most of the sheeting wood on there only to give it shape?
This part of your comment jumps out at me. The sheeting on the tail surfaces is where they get their strength. If memory serves me the sheeting on the tail surfaces is 1/16", easily sanded through if not careful. If you've sanded too much at the root of the stabilizer then it will need to be reinforced. Either by laminating another piece of balsa sheet over it or by laminating fiberglass in the area where it has been weakened. The horizontal stabilizer is under a great deal of stress during certain maneuvers such as pulling out at the bottom of a steep/fast dive. If it's weakened it will break sooner or later. I've taken my plane up high and dove it back down to pull out at 40 or 50 ft. If the plane is close enough you can actually see the stabilizer flex under the load.